We observe that if there is any secant \(({\mathbf {x}},{\mathbf {x}}^{\prime })\in \mathcal {X}_N\times \mathcal {X}_N\)  for which Eq.  REF is not satisfied for a given \({S}\subset {M}\) , then the corresponding term of Eq.  REF is less than \(1/L^2\)  and \(f_{L}({S})<f_{L}({M})\) .
Otherwise, each term of Eq.  REF is \(1/L^2\)  and we have \(f_{L}({S})=f_{L}({M})\) .
Again, the classical result in [1]} shows that a greedy approximation algorithm maximizing Eq.  REF and stopping when \(f_{L}({S}_K) = f_{L}({M})\)  finds the minimum possible number of sensors up to a logarithmic factor so that the Lipschitz condition Eq.  REF is satisfied.
In particular, the same guarantee stated in Eq.  REF holds for the Lipschitz objective too.
